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Traversals A systematic method to visit all nodes in a tree Binary tree traversals: Pre-order: root, left, right In-order: left, root, right Post-order: left, right, root General graph traversals (searches) Depth-first search Breadth-first search

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Inorder(tree t) 1. if t = nil 2. return 3. inorder(t.left) 4. visit(t) // e.g., print it 5. inorder(t.right)

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Inorder (Infix) 1 2 3 5 6 7 8 9 10 (a BST will always work well with in-order traversal) 5213861079

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Pre-order (Prefix) 5 2 1 3 8 6 7 10 9 5213861079

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Post-order (Postfix) 1 3 2 7 6 9 10 8 5 5213861079

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Post-order 1 3 * + 6 10 - +*13-610

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Depth-first search (DFS) חיפוש לעומק)) DFS: Search one subtree completely before other Pre-order traversal is an example of a DFS: Visit root, left subtree (all the way), visit right subtree We can do it in other order: Visit root, right subtree, left subtree

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Depth-first search (DFS) DFS: visit all descendents before siblings 5213861079 5 2 1 3 8 6 7 10 9

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DFS(tree t) 1. q new stack 2. push(q, t) 3. while (not empty(q)) 4. curr pop(q) 5. visit curr // e.g., print curr.datum 6. push(q, curr.left) 7. push(q, curr.right) This version for binary trees only!

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Breadth-first search (BFS) (חיפוש לרוחב) BFS: visit all siblings before their descendents 5213861079 5 2 8 1 3 6 10 7 9

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BFS(tree t) 1. q new queue 2. enqueue(q, t) 3. while (not empty(q)) 4. curr dequeue(q) 5. visit curr // e.g., print curr.datum 6. enqueue(q, curr.left) 7. enqueue(q, curr.right) This version for binary trees only!

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DFS(tree t) 1. q new stack 2. push(q, t) 3. while (not empty(q)) 4. curr pop(q) 5. visit curr // e.g., print curr.datum 6. push(q, curr.left) 7. push(q, curr.right) This version for binary trees only!

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DFS(tree t) Void Graph::dfs (Vertex v) { v.visted = true; for each w adjacent to v if (!w.visited) dfs(w); } This version for any type of trees (graph)

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Graphs vs. Trees Graphs don’t have any root Graphs can be directed or undirected Trees only grow down (upside-down) (Why do trees grow upside down for Computer scientists???) Graphs can have cycles, trees can’t (why?)

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DFS Example on Graph source vertex

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AVL (Adelson, Velskii, Landis) Balance the tree (not our targil) The left and right branches of the tree can only differ by one level Ensures log N depth (much better for searching) Takes log N to add or delete

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5 38 1 410 5 3 1 4 AVL Tree Not AVL Tree

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AVL Trees Trees often need to be “rotated” when deleting or inserting to keep AVL balance Nice link on this process Nice link on this process Sample AVL code Sample AVL code

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